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Quantum Physics

Title:Quantum Marginal Problem and its Physical Relevance

Abstract: The Pauli exclusion principle as constraint on fermionic occupation numbers
is a consequence of the much deeper fermionic exchange statistics. Just
recently, it was shown by Klyachko that this antisymmetry of fermionic wave
functions leads to further restrictions on natural occupation numbers. These
so-called generalized Pauli constraints (GPC) significantly strengthen Pauli's
exclusion principle. Our first goal is to develop an understanding of the
mathematical concepts behind Klyachko's work, in the context of quantum
marginal problems. Afterwards, we explore the physical relevance of GPC and
study concrete physical systems from that new viewpoint.
In the first part of this thesis we review Klyachko's solution of the
univariate quantum marginal problem. In particular we break his abstract
derivation based on algebraic topology down to a more elementary level and
reveal the geometrical picture behind it.
The second part explores the possible physical relevance of GPC. We review
the effect of pinning, i.e. the saturation of some GPC by given natural
occupation numbers and explain its consequences. Although this effect would be
quite spectacular we argue that pinning is unnatural. Instead, we conjecture
the effect of quasipinning, defined by occupation numbers close to (but not
exactly on) the boundary of the allowed region.
In the third part we study concrete fermionic quantum systems from the new
viewpoint of GPC. In particular, we compute the natural occupation numbers for
the ground state of a family of interacting fermions in a harmonic potential.
Intriguingly, we find that the occupation numbers are strongly quasipinned,
even up to medium interaction strengths. We identify this as an effect of the
lowest few energy eigenstates, which provides first insights into the mechanism
behind quasipinning.